QCD corrections to charged-current decays with Heavy Sterile Neutrinos in initial or final state and their impact on $τ$ decays

Searches for a Heavy Sterile Neutrino $N$ profit from precise predictions of inclusive decay rates, entering predictions for branching fractions and lifetime. Once decay channels into semi-hadronic final states are open, a reliable calculation of inclusive decay rates is only possible if $N$ is heavy enough to permit a perturbative calculation. We adopt the scenario in which $N$ only interacts with SM particles through $N$-$ν_\ell$ mixing, where $\ell=e,μ,τ$. Using literature results for $W$ boson correlators calculated to $\mathcal{O}(α_s^4)$, we study the quality of the perturbation series for $N\to \ell +\mbox{hadrons}$ to determine mass ranges for which inclusive decay widths can be predicted robustly. We present novel analytic results for the decay rate $N\to τ+\mbox{hadrons}$ in terms of $m_τ/m_N$. Our expressions equally apply to $τ\to N +\mbox{hadrons}$, perturbatively calculable for $m_N\lesssim 600\,$MeV. Applying our result to the $τ$ lifetime, we determine the allowed parameter space for the $N$-$ν_τ$ mixing angle $θ$ and $m_N$. We find $|\sinθ| \leq 0.2 $ for $m_N=600\,$MeV and weaker bounds for a lighter $N$. For $m_N\geq m_τ$ we find constraints from the dependence of $τ$ decay rates on $\cosθ$. ... mehrCombining $τ\to π^- ν_τ$ and $τ\to K^- ν_τ$ data gives $|\sinθ| = (9.09 \pm 3.56) \cdot 10^{-2}$ while $N$-$ν_τ$ mixing does not improve the agreement between theory and data for $τ\to \ell \barν_\ell ν_τ$. We find current data for the decay rate $Γ(τ\to \ell+\mbox{nothing})$ about 1$σ$ above the SM prediction for $Γ(τ\to \ell \barν_\ell ν_τ)$, which leads to useful constraints on $Γ(τ\to \ell X_{\mathrm{dark}})$ with dark-sector particles $X_{\mathrm{dark}}$ and might stimulate additional experimental effort on $τ\to \ell+\mbox{nothing}$.